week ending PRL97,133902(20 PHYSICAL REVIEW LETTERS 29 SEPTEMBER 2006 No. FA8721-05-C-0002, and by the Chinese National Foundation under Contract Nos. 60371010 and No. 60531020. Opinions, interpretations, conclusions, and recommendations are those of the author and are not necessarily endorsed by the United States Government [1 H. Minkowski, Nachr. Akad. Wiss. Gottingen 1, 53 2000 (1908). diameter [nm [2 M. Abraham, Rend. Circ. Mat. Palermo 28, 1(1909) [3] D F. Nelson, Phys. Rev. A 44, 3985(1991) FIG4. Force versus diameter for a dielectric sphere(E/Eo= [4] J.A. Kong, Electromagnetic Wave Theory (EMW 16+ i)incident by a unit amplitude plane wave. The free-space Publishing. Cambridge. MA. 2005 wavelength of the incident wave is o =1064 nm. 5]J.P. Gordon, Phys. Rev. A 8, 14(1973) [6 R. Loudon, Fortschr. Phys. 52, 1134(2004) [7]R Loudon, J Mod. Opt. 49, 821(2002) Fig 3(b)due to absorption. This separation of F into Fb [81 Y.N. Obukhov and F.w. Hehl, Phys. Lett. A 311, 277 and Fc is further investigated by plotting the force versus (2003) sphere diameter for constant material parameters in Fig. 4. [9] M. Mansuripur, Opt. Express 12, 5375(2004) For small spheres, the power absorption is small since the [10] B.A. Kemp, T M. Grzegorczyk, and J.A.Kong, Opt diameter is much less than the penetration depth. When the Express13.9280(2005) diameter is of the order of the penetration depth, the force [I] B.A. Kemp, T.M. Grzegorczyk, and J.A. Kong, on free currents becomes significant due to the direct J. Electromagn. Waves. Appl. 20, 827(2006 dependence upon n given by (4) [12] P. Penfield and H. A. Haus, Electrodynamics of Moving In this Letter, we have provided a perspective of mo- Media (m.I.t. Press, Cambridge, MA, 1967) mentum transfer in lossy media in agreement with the [13]R. Loudon, L. Allen, and D F. Nelson, Phys. Rev. E55, distribution of Lorentz force and relevant experiments. In 071(1997). the case of an absorbing Mie particle, the contributions [14]S. Stallinga, Phys. Rev. E 73. 026606(2006 from Fh and Fc sum to give the total force on the particle. [15] R Loudon, S M. Barnett, and C. Baxter, Phys. Rev.a71 063802(2005) The particles we consider consist of ER-16Eo, a value [16] G K. Campbell, A.E. Leanhardt, J. Mun, M. Boyd, EW typical for semiconductors, and ER= 2, which is repre treed, w. Ketterle, and D. E. Pritchard, Phys. Rev. Lett. sentative of many insulators. A novelty of our results is the 94,170403(2005) reduction of optical momentum transfer to particles due to [17] A.E. Gibson, M.E. Kimmitt, and A C. Walker, Appl bsorption, which requires high dielectric contrast with the Phys.Let.17,75(1970). background medium and an attenuation length on the order [18] M. Mansuripur, Opt. Express 13, 2245(2005) of particle diameter. These results differ from the expected [19] J.A. Stratton, Electromagnetic Theory(McGraw-Hill, esult of scattering plus absorption forces resulting from New York, 1941) Rayleigh particles [25]. Because a detailed understanding [21] R V Jones and J.C. S Richards, ProcRSoc. A221,480 of both Fb and Fc are required to describe the physics involved, the theory presented here is fundamental to the (221 R.V. Jones and B. Leslie, Proc R Soc. A 360, 347(1978) understanding of optical momentum transfer to absorbing [23] T M. Grzegorczyk,BA. Kemp, and J.A.Kong,J.Opt particles Soc.Am.A23,2324(2006) This work is sponsored by NASA-USRA under [24] T.M. Grzegorczyk, B A. Kemp, and J.A. Kong, Phys Contracts No. NAS5-03110 and No. 07605-003-055, by Rev.Let.96,113903(2006) the Department of the Air Force under Contract [25]K Svoboda and S.M. Block, Opt Lett. 19, 930(1994) 133902-4Fig. 3(b) due to absorption. This separation of F into Fb and Fc is further investigated by plotting the force versus sphere diameter for constant material parameters in Fig. 4. For small spheres, the power absorption is small since the diameter is much less than the penetration depth. When the diameter is of the order of the penetration depth, the force on free currents becomes significant due to the direct dependence upon n given by (4). In this Letter, we have provided a perspective of mo￾mentum transfer in lossy media in agreement with the distribution of Lorentz force and relevant experiments. In the case of an absorbing Mie particle, the contributions from F
b and F
c sum to give the total force on the particle. The particles we consider consist of
R
16
0, a value typical for semiconductors, and
R
2, which is repre￾sentative of many insulators. A novelty of our results is the reduction of optical momentum transfer to particles due to absorption, which requires high dielectric contrast with the background medium and an attenuation length on the order of particle diameter. These results differ from the expected result of scattering plus absorption forces resulting from Rayleigh particles [25]. Because a detailed understanding of both F
b and F
c are required to describe the physics involved, the theory presented here is fundamental to the understanding of optical momentum transfer to absorbing particles. This work is sponsored by NASA-USRA under Contracts No. NAS5-03110 and No. 07605-003-055, by the Department of the Air Force under Contract No. FA8721-05-C-0002, and by the Chinese National Foundation under Contract Nos. 60371010 and No. 60531020. Opinions, interpretations, conclusions, and recommendations are those of the author and are not necessarily endorsed by the United States Government. *Electronic address: bkemp@mit.edu [1] H. Minkowski, Nachr. Akad. Wiss. Go¨ttingen 1, 53 (1908). [2] M. Abraham, Rend. Circ. Mat. Palermo 28, 1 (1909). [3] D. F. Nelson, Phys. Rev. A 44, 3985 (1991). [4] J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, Cambridge, MA, 2005). [5] J. P. Gordon, Phys. Rev. A 8, 14 (1973). [6] R. Loudon, Fortschr. Phys. 52, 1134 (2004). [7] R. Loudon, J. Mod. Opt. 49, 821 (2002). [8] Y. N. Obukhov and F. W. Hehl, Phys. Lett. A 311, 277 (2003). [9] M. Mansuripur, Opt. Express 12, 5375 (2004). [10] B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, Opt. Express 13, 9280 (2005). [11] B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, J. Electromagn. Waves. Appl. 20, 827 (2006). [12] P. Penfield and H. A. Haus, Electrodynamics of Moving Media (M.I.T. Press, Cambridge, MA, 1967). [13] R. Loudon, L. Allen, and D. F. Nelson, Phys. Rev. E 55, 1071 (1997). [14] S. Stallinga, Phys. Rev. E 73, 026606 (2006). [15] R. Loudon, S. M. Barnett, and C. Baxter, Phys. Rev. A 71, 063802 (2005). [16] G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, Phys. Rev. Lett. 94, 170403 (2005). [17] A. F. Gibson, M. F. Kimmitt, and A. C. Walker, Appl. Phys. Lett. 17, 75 (1970). [18] M. Mansuripur, Opt. Express 13, 2245 (2005). [19] J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). [20] J. H. Poynting, Philos. Mag. 9, 169 (1905). [21] R. V. Jones and J. C. S. Richards, Proc. R. Soc. A 221, 480 (1954). [22] R. V. Jones and B. Leslie, Proc. R. Soc. A 360, 347 (1978). [23] T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, J. Opt. Soc. Am. A 23, 2324 (2006). [24] T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, Phys. Rev. Lett. 96, 113903 (2006). [25] K. Svoboda and S. M. Block, Opt. Lett. 19, 930 (1994). 0 500 1000 1500 2000 −1 0 1 2 3x 10−23 diameter [nm] F [N] F Fc Fb FIG. 4. Force versus diameter for a dielectric sphere (
=
0
16 i) incident by a unit amplitude plane wave. The free-space wavelength of the incident wave is 0
1064 nm. PRL 97, 133902 (2006) PHYSICAL REVIEW LETTERS week ending 29 SEPTEMBER 2006 133902-4